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GEOMETRIC ORBITAL INTEGRALS AND THE
CENTER OF THE ENVELOPING ALGEBRA
Jean-Michel Bismut, Shu Shen
To cite this version:
Jean-Michel Bismut, Shu Shen. GEOMETRIC ORBITAL INTEGRALS AND THE CENTER OF THE ENVELOPING ALGEBRA. 2019. �hal-02332185�
CENTER OF THE ENVELOPING ALGEBRA
JEAN-MICHEL BISMUT AND SHU SHEN
Abstract. The purpose of this paper is to extend the explicit geo-metric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.
Contents
1. Introduction 3
2. Geometric formulas for orbital integrals and the Casimir 5
2.1. Reductive groups and symmetric spaces 6
2.2. Semisimple elements and their displacement function 7
2.3. Enveloping algebra and the Casimir 10
2.4. The elliptic operator Cg,X 12
2.5. Orbital integrals and the Casimir 13
2.6. The function Jγ 15
2.7. Some properties of the function Jγ 16
2.8. A geometric formula for the orbital integrals associated
with the Casimir 17
3. Cartan subalgebras, Cartan subgroups, and root systems 19
3.1. Linear algebra 19
3.2. The Cartan subalgebras of g 20
3.3. A root system and the Weyl group 21
3.4. Real roots and imaginary roots 23
3.5. A positive root system 25
3.6. The case when h is fundamental and the root system of
(hk, k) 27
3.7. Cartan subgroups and regular elements 28
2010 Mathematics Subject Classification. 11F72, 22E30,
Key words and phrases. Selberg trace formula, Analysis on real and complex Lie groups.
The authors are much indebted to Laurent Clozel for his stimulating remarks during the preparation of the paper, and for reading the preliminary version of this paper very carefully.
3.8. Cartan subgroups and semisimple elements 29
3.9. Root systems and their characters 30
3.10. Real roots, imaginary roots, and semisimple elements 31
3.11. Cartan subalgebras and differential operators 32
4. Root systems and the function Jγ 34
4.1. The determinant of 1 − Ad (γ) 34
4.2. Evaluation of the function Jγ on ihk 38
5. The function Jγ when γ is regular 41
5.1. A neighborhood of γ in H 41
5.2. The γ-regular elements in h 42
5.3. The function DH(γ) 43
5.4. The function Jγ when γ is regular 43
6. The Harish-Chandra isomorphism 44
6.1. The center of the enveloping algebra 45
6.2. The complex form of the Harish-Chandra isomorphism 46
6.3. The real form of the Harish-Chandra isomorphism 47
6.4. The Duflo and the Harish-Chandra isomorphisms 49
6.5. The case of the Casimir 49
6.6. The action of Z (g) on C∞(X, F ) 50
6.7. The semisimple orbital integrals involving Z (g) 51
7. The center of U (g) and the regular orbital integrals 52
7.1. The algebra Z (g) and the regular orbital integrals 52
7.2. A geometric formula for the regular orbital integrals 53
8. The function Jγ and the limit of regular orbital integrals 55
8.1. The function Jγ when γ is not regular 55
8.2. The Lie algebra z (γ) and the isomorphisms of
Harish-Chandra and Duflo 60
8.3. An application of Rossmann’s formula 61
8.4. The limit of certain orbital integrals 63
9. The final formula 66
9.1. The general case 67
9.2. A microlocal version 68
9.3. Compatibility properties of the formula 70
10. Orbital integrals and the index theorem 71
10.1. The Dirac operator on X 71
10.2. The case where dim b = 0 73
10.3. Orbital integrals and the index theorem: the case of the
identity 73
10.4. The case where γ = k−1, dim b = 0 77
10.5. Orbital integrals and index theory: the case of elliptic
elements 78
Index 85
References 87
1. Introduction
In [B11, Chapter 6], one of us established a geometric formula for the semisimple orbital integrals of smooth kernels associated with the Casimir. The purpose of this paper is to extend this formula to the smooth kernels where more general elements of the center of the en-veloping algebra also appear.
Let us briefly describe our main result in more detail. Let G be a connected real reductive group, and let g be its Lie algebra. Let θ ∈ Aut (G) be a Cartan involution, and let K ⊂ G be the corresponding maximal compact subgroup with Lie algebra k. Let g = p ⊕ k be the associated Cartan splitting. Let B be a symmetric nondegenerate bilinear form on g which is G and θ invariant, positive on p and negative on k. Let X = G/K be the associated symmetric space, a Riemannian manifold with parallel nonpositive curvature.
Let ρE : K → U (E) be a finite dimensional unitary representation of
K, and let F = G×KE be the corresponding vector bundle on X. Then
G acts on the left on C∞(X, F ). Let U (g) be the enveloping algebra of g, and let Z (g) be the center of U (g). Then Z (g) acts on C∞(X, F ) and its action commutes with the left action of G. Among the elements of Z (g), there is the Casimir Cg, whose action on C∞(X, F ) is denoted Cg,X.
Let Seven(R) denote the even real functions on R that lie in the
Schwartz space S (R). Let µ ∈ Seven(R) be such that if
b
µ ∈ Seven(R)
is its Fourier transform, there is C > 0, and for any k ∈ N, there is ck> 0 such that
(1.1) µb(k)(y)
≤ ckexp −Cy2 .
If A ∈ R, µ√Cg,X + A is a well-defined operator with a smooth
kernel.
If γ ∈ G is semisimple, as explained in [B11, Section 6.2], the orbital integral Tr[γ]hµ√Cg,X + Ai is well-defined, and it only depends on
the conjugacy class of γ in G. After conjugation, we can write γ in the form γ = eak−1, a ∈ p, k ∈ K, Ad (k−1) a = a. If Z (γ) ⊂ G is the centralizer of γ with Lie algebra z (γ), then θ acts on Z (γ), and Z (γ) is a possibly nonconnected reductive group. Let z (γ) = p (γ) ⊕ k (γ) be the associated Cartan splitting.
Let I·(g) be the algebra of invariant polynomials on g∗, and let τD:
I·(g) ' Z (g) denote the Duflo isomorphism [Du70, Thorme V.2]. If h⊂ g is a Cartan subalgebra, let I·(h, g) denote the algebra of
polyno-mials on h∗ that are invariant under the corresponding algebraic Weyl group, so that we have the canonical identification I·(g) ' I·(h, g)1. There is a Harish-Chandra isomorphism φHC : Z (g) ' I·(h, g). By
[Du70, Lemme V.1], the Duflo and Harish-Chandra isomorphisms are known to be compatible.
There is a canonical projection g → z (γ)2, that induces a corre-sponding projection I·(g) → I·(z (γ)). If L ∈ Z (g), let Lz(γ) denote
the differential operator on z (γ) canonically associated with the projec-tion of τD−1L on I·(z (γ)). In particular, up to a constant, − (Cg)z(γ)
ex-tends to the standard Laplacian on the Euclidean vector space zi(γ) =
p(γ) ⊕ ik (γ).
Following [B11, Chapter 5], in Definition 2.6, we define a smooth function Jγ : ik (γ) → C. Let δa be the Dirac mass at a ∈ p (γ). Then
Jγ Y0k Tr EhρEk−1e−Yk 0 i ⊗ δa is a distribution on zi(γ). 3
Our main result, which is repeated in the text as Theorem 9.1 is as follows.
Theorem 1.1. The following identity holds: (1.2) Tr[γ]hLµpCg,X+ Ai = Lz(γ)µ q (Cg)z(γ)+ A h Jγ Y0k Tr Eh ρEk−1e−Y0k i δa i (0) . When L = 1, our theorem was already established in [B11, Theorem 6.2.2].
The proofs in [B11] used a construction of a new object, the hypoel-liptic Laplacian. Here, we will only need the results of [B11].
Our proof is done in two steps. In a first step, using the results of [B11], we prove Theorem1.1 when γ ∈ G is regular. In this case, using the properties of the Harish-Chandra isomorphism [HC66], the proof is relatively easy.
1This isomorphism is usually written in its complex version I·(g
C) ' I·(hC, gC).
In Subsections3.3 and6.3, the corresponding real version is derived. Such consid-erations will also apply to other complex isomorphisms.
2This projection is defined in Subsection8.2. 3In the sequel, ⊗ will be omitted.
When γ is nonregular, we combine our result for γ regular with limit arguments due to Harish-Chandra on the behavior of orbital integrals when γ0 regular converges to γ. In both steps, remarkable and non-trivial properties of the function Jγ are used. One of these properties
is that one essential piece of Jγ can be calculated only in terms of
imaginary roots.
This paper is organized as follows. In Section 2, we describe the geometric setting, and we explain the formula for the semisimple orbital integrals that was obtained in [B11].
In Section3, we recall some of the properties of Cartan subalgebras, Cartan subgroups, and of the corresponding root systems.
In Section 4, we express the restriction of the function Jγ to Cartan
subalgebras in terms of a positive root system.
In Section 5, we specialize the results of the previous section to the case where γ is regular. We prove a crucial and unexpected smooth dependence of Jγ on γ.
In Section 6, we explain in some detail the Harish-Chandra isomor-phism.
In Section7, we establish Theorem 1.1 when γ is regular.
In Section 8, when γ is non necessarily regular, we study the limit of Jγ0, and the limit of our formula for regular orbital integrals as γ0
regular converges to γ in a suitable sense.
In Section 9, using the results of the previous section, we establish Theorem 1.1 in full generality.
Finally, in Section 10, we prove that our formula is compatible to the index theory for Dirac operators, and also with known results on Dirac cohomology [HuP02].
2. Geometric formulas for orbital integrals and the Casimir
In this Section, we explain the geometric formula given in [B11, Chapter 6] for the semisimple orbital integrals associated with the proper smooth kernels for the Casimir.
This section is organized as follows. In Subsection2.1, we introduce the real reductive group G, its maximal compact subgroup K, the Lie algebras g, k, and the symmetric space X = G/K.
In Subsection 2.2, we recall the definition of semisimple elements in G, and of the corresponding displacement function.
In Subsection 2.3, we introduce the enveloping algebra U (g), and the Casimir element Cg ∈ U (g).
In Subsection2.4, given a unitary representation of K, we construct the corresponding vector bundle F on X, and the elliptic operator Cg,X
which is just the action of Cg on C∞(X, F ).
In Subsection2.5, given µ ∈ Seven(R) such that its Fourier transform
has the proper Gaussian decay, if A ∈ R, we recall the definition of the semisimple orbital integrals associated with the smooth kernel for µ√Cg,X + A. Among these kernels, there is the heat kernel for Cg,X.
In Subsection 2.6, if γ ∈ G is semisimple, if Z (γ) ⊂ G is its cen-tralizer with Lie algebra z (γ), if k (γ) is the compact part of z (γ), we recall the definition of the function Jγ on ik (γ) given in [B11, Theorem
5.5.1].
In Subsection 2.7, we study the behavior of Jγ when replacing by γ
by γ−1, and also by complex conjugation.
Finally, in Subsection 2.8, we state the geometric formula obtained in [B11] for the above orbital integrals, in which the function Jγ plays
a key role.
2.1. Reductive groups and symmetric spaces. Let G be a con-nected reductive real Lie group, and let g be its Lie algebra. Let θ ∈ Aut (G) be a Cartan involution. Then θ acts as an automorphism of g. Let K ⊂ G be the fixed point set of θ. Then K is a compact connected subgroup of G, which is a maximal compact subgroup. If k ⊂ g is the Lie algebra of K, then k is the fixed point set of θ in g. Let p ⊂ g be the eigenspace of θ corresponding to the eigenvalue −1, so that we have the Cartan decomposition
(2.1) g= p ⊕ k. Put m = dim p, n = dim k, (2.2) so that (2.3) dim g = m + n.
Let B be a G and θ invariant bilinear symmetric nondegenerate form on g. Then (2.1) is a B-orthogonal splitting. We assume that B is positive on p and negative on k. Let h i = −B (·, θ·) be the corresponding scalar product on g. Let B∗ be the bilinear symmetric form on g∗ = p∗⊕ k∗ which is dual to B.
Let ωg be the canonical left-invariant 1-form on G with values in g.
By (2.1), ωg splits as
Let X = G/K be the corresponding symmetric space. Then p : G → X = G/K is a K-principal bundle, and ωk is a connection form. Also
the tangent bundle T X is given by
(2.5) T X = G ×Kp.
Then T X is equipped with the scalar product h i induced by B, so that X is a Riemannian manifold. The connection ∇T X on T X which is
induced by ωk is the Levi-Civita connection of T X, and its curvature
is parallel and nonpositive. Also G acts isometrically on the left on X, and θ acts as an isometry of X.
By [Kn86, Proposition 1.2], any element γ ∈ G factorizes uniquely in the form
γ = eak−1, a ∈ p, k ∈ K.
(2.6)
If γ, g ∈ G, set
(2.7) C (g) γ = gγg−1.
Then C (g) is an automorphism of G. Its derivative at the identity is the adjoint representation g ∈ G → Ad (g) ∈ Aut (g). The derivative of this last map is given by a ∈ g → ad (a) ∈ End (g), with ad (a) b = [a, b]. If γ ∈ G, the fixed point set of C (γ) is the centralizer Z (γ) ⊂ G, whose Lie algebra z (γ) is given by
(2.8) z(γ) = ker (1 − Ad (γ)) .
If f ∈ g, let Z (f ) ⊂ G be the stabilizer of f . Its Lie algebra z (f ) ⊂ g is given by
(2.9) z(f ) = ker ad (f ) .
In the sequel, if M is a Lie group, we denote by M0 the connected component of the identity.
2.2. Semisimple elements and their displacement function. Let d be the Riemannian distance on X. By [BaGS85, 6.1], d is a convex function on X × X. If γ ∈ G, let dγ be the corresponding displacement
function on X, i.e., (2.10) dγ(x) = d (x, γx) . If g ∈ G, then (2.11) dC(g)γ(gx) = dγ(x) . Moreover, (2.12) dθ(γ)(θx) = dγ(x) .
Set
(2.13) mγ = inf dγ.
Let X (γ) ⊂ X be the closed subset where dγ reaches its minimum. By
[BaGS85, p. 78 and 1.2], X (γ) is a closed convex subset, dγ is smooth
on X \ X (γ) and has no critical points on X \ X (γ). Also by (2.11), (2.12),
X (C (g) γ) = gX (γ) , X (θγ) = θX (γ) ,
(2.14)
mC(g)γ = mγ, mθ(γ) = mγ.
By [E96, Definition 2.19.21], γ is said to be semisimple if X (γ) is nonempty. If γ is semisimple, then C (g) γ and θ (γ) are semisimple. Also γ is said to be elliptic if it is semisimple and mγ = 0. Elliptic
elements are exactly the group elements that are conjugate to elements of K. Finally, γ is said to be hyperbolic if it is conjugate to ea, a ∈ p.
By [Ko73, Proposition 2.1], [BaGS85, Theorems 2.19.23 and 2.19.24], γ ∈ G is semisimple if and only if it factorizes as γ = he = eh, with commuting hyperbolic h and elliptic e. Also e, h are uniquely determined by γ, and
(2.15) Z (γ) = Z (h) ∩ Z (e) .
Set
(2.16) x0 = p1.
Theorem 2.1. Let γ ∈ G be semisimple. If g ∈ G, x = pg ∈ X, then x ∈ X (γ) if and only if there exist a ∈ p, k ∈ K such that Ad (k) a = a, and
(2.17) γ = C (g) eak−1 .
Also C (g) ea ∈ G, C (g) k ∈ G are uniquely determined by γ. If g t =
geta, then t ∈ [0, 1] → y
t = pgt is the unique geodesic connecting x and
γx. Moreover
(2.18) mγ = |a| .
If γ ∈ G is semisimple, then x0 ∈ X (γ) if and only if there exist
a ∈ p, k ∈ K such that
γ = eak−1, a ∈ p, Ad (k) a = a.
(2.19)
Also a, k are uniquely determined by (2.19).
Proof. The first part of our theorem was established in [B11, Theorem 3.1.2]. By taking g = 1 in the first part, we obtain the second part.
Let γ ∈ G be a semisimple element written as in (2.19). By [B11, Proposition 3.2.8 and eqs. (3.3.4), (3.3.6)],
Z (ea) = Z (a) , Z (γ) = Z (a) ∩ Z (k) , z(γ) = z (a) ∩ z (k) . (2.20)
By (2.19), a ∈ z (γ), and by (2.20), z (γ) ⊂ z (a), so that a is an element of the center of z (γ).
Clearly,
(2.21) θ (γ) = e−ak−1.
Therefore θ (γ) ∈ Z (γ), so that the above centralizers and Lie algebras are preserved by θ. Set
(2.22) K (γ) = Z (γ) ∩ K.
By [B11, Theorem 3.3.1], we have the identity
(2.23) K0(γ) = Z0(γ) ∩ K,
and K0(γ) is a maximal compact subgroup of Z0(γ). Put
p(γ) = p ∩ z (γ) , k(γ) = k ∩ z (γ) . (2.24)
Then k (γ) is the Lie algebra of K (γ). We use similar notation for the Lie algebras z (k) , z (a). We have the Cartan decompositions of Lie algebras,
z(γ) = p (γ) ⊕ k (γ) , z (k) = p (k) ⊕ k (k) , z(a) = p (a) ⊕ k (a) . (2.25)
Then B restricts to a nondegenerate form on z (γ) , z (k) , z (a), so that Z (γ) , Z (k) , Z (a) are possibly nonconnected reductive subgroups of G. By [B11, Theorem 3.3.1], we have the identification of finite groups,
(2.26) Z0(γ) \ Z (γ) = K0(γ) \ K (γ) .
Let z⊥(γ) , z⊥(a) be the orthogonal spaces to z (γ) , z (a) in g with respect to B. We have splittings
z⊥(γ) = p⊥(γ) ⊕ k⊥(γ) , z⊥(a) = p⊥(a) ⊕ k⊥(a) . (2.27)
Let z⊥a (γ) denote the orthogonal to z (γ) in z (a). We still have a splitting
(2.28) z⊥a (γ) = p⊥a (γ) ⊕ k⊥a (γ) .
Theorem 2.2. The set X (γ) is preserved by θ. Moreover,
(2.29) X (γ) = X (ea) ∩ X (k) .
Also X (γ) is a totally geodesic submanifold of X. In the geodesic coordinate system centered at x0 = p1, then
(2.30) X (γ) = p (γ) .
The actions of Z0(γ) , Z (γ) on X (γ) are transitive. More precisely
the maps g ∈ Z0(γ) → pg ∈ X, g ∈ Z (γ) → pg ∈ X induce the
identification of Z0(γ)-manifolds,
(2.31) X (γ) = Z0(γ) /K0(γ) = Z (γ) /K (γ) .
Now we will establish a simple important consequence of Theorem
2.2.
Theorem 2.3. Let γ be a semisimple element of G as in (2.19). Let γ0 be another semisimple element of G such that
γ0 = ea0k0−1, a0 ∈ p, Ad (k0) a0 = a0. (2.32)
Then there exists g ∈ G such that γ0 = C (g) γ if and only if there exists k00∈ K such that C (k00) γ = γ0, in which case
a0 = Ad (k00) a, k0 = C (k00) k. (2.33)
Proof. Assume that γ0 = C (g) γ. By (2.14), we get
(2.34) X (γ0) = gX (γ) .
By Theorem 2.1, x0 ∈ X (γ) ∩ X (γ0). By (2.34), gx0 ∈ X (γ0). By
Theorem 2.2, there exists h ∈ Z (γ0) such that
(2.35) hgx0 = x0,
which is equivalent to
(2.36) k00= hg ∈ K.
Since h ∈ Z (γ0), we conclude that C (k00) γ = γ0. Using the uniqueness of decomposition in (2.32), equation (2.33) follows. The proof of our
theorem is completed.
2.3. Enveloping algebra and the Casimir. We identify g with the Lie algebra of left-invariant vector fields on G. Let U (g) be the en-veloping algebra of g. Then U (g) can be identified with the algebra of left-invariant differential operators on G. Let Z (g) ⊂ U (g) denote the center of U (g).
If E is a finite dimensional real or complex vector space, and if ρE : g → End (E) is a morphism of Lie algebras, the map ρE extends
Among the elements of Z (g), there is the Casimir element Cg. If e1, . . . , em+n is a basis of g, and if e∗1, . . . , e
∗
m+n is the dual basis of g
with respect to B, then 4
(2.37) Cg= −
m+n
X
i=1
e∗iei.
If we consider instead the Lie algebra (k, B|k), Ck ∈ Z (k) denotes the
associated Casimir element.
If e1, . . . , em is a basis of p, and if e∗1, . . . , e∗m is the dual basis of p
with respect to B|p, set
(2.38) Cp = −
m
X
i=1
e∗iei.
Then Cp∈ U (g). Using (2.37), (2.38), we get
(2.39) Cg= Cp+ Ck.
Also Cp and Ck commute.
If ρE : g → End (E) is taken as above, put
(2.40) Cg,E = ρE(Cg) .
Under the above conditions, we can define Cp,E, Ck,E.
Since g is itself a representation of g, Cg,g is the action of Cg on g.
Since k acts on p, k, Ck,p, Ck,k are also well-defined.
Proposition 2.4. The following identity holds: (2.41) Tr [Cg,g] = 3TrCk,p + Tr Ck,k .
Proof. By (2.39), we get
(2.42) Tr [Cg,g] = Tr [Cp,g] + TrCk,g .
Let e1, . . . , em be an orthonormal basis of p, and let em+1, . . . , en be
an orthonormal basis of k. Then Tr [Cp,g] = − X 1≤i≤m 1≤j≤m+n |[ei, ej]|2, (2.43) TrCk,p = − X 1≤i≤m m+1≤j≤m+n |[ei, ej]|2 = − X 1≤i,j≤m |[ei, ej]|2.
4In [Kn86, Section 8.3], the Casimir is defined with the opposite sign. We have
By (2.43), we deduce that
(2.44) Tr [Cp,g] = 2TrCk,p .
Also
(2.45) TrCk,g = Tr Ck,p + Tr Ck,k .
By (2.42), (2.44), and (2.45), we get (2.41). The proof of our
proposi-tion is completed.
Let h ⊂ g be a θ-stable Cartan subalgebra. Then we have the split-ting h = hp⊕ hk. 5 Let R ⊂ h∗C be the corresponding root system. Let
R+ be a positive root system. If R− = −R+, R is the disjoint union
of R+ and R−. Let ρg ∈ h∗C be the half sum of the positive roots.
Here ρg ∈ h∗ p⊕ ih
∗
k. By Kostant’s strange formula [Ko76], we have the
identity (2.46) B∗(ρg, ρg) = − 1 24Tr [C g,g] . By (2.41), (2.46), we get (2.47) B∗(ρg, ρg) = −1 8TrC k,p − 1 24TrC k,k .
Using the notation in [B11, eq. (2.6.11)] 6, by (2.47), we obtain
(2.48) B∗(ρg, ρg) = −1
4B
∗
(κg, κg) .
2.4. The elliptic operator Cg,X. Let E be a finite dimensional Her-mitian vector space, and let ρE : K → U (E) denote a unitary repre-sentation of K. The Casimir Ck,E is a self-adjoint nonpositive
endo-morphism of E. If ρE is irreducible, then Ck,E is a scalar.
Let F be the vector bundle on X,
(2.49) F = G ×KE.
Then F is a Hermitian vector bundle on X, which is equipped with a canonical connection ∇F. Also Ck,E descend to a parallel section Ck,F of End (F ). Moreover, G acts on C∞(X, F ), so that if g ∈ G, s ∈ C∞(X, F ), if g∗ denotes the lift of the action of g to F ,
(2.50) gs (x) = g∗s g−1x .
The Casimir operator Cg descends to a second order elliptic operator
Cg,X acting on C∞(X, F ), which commutes with G. Let ∆X denote 5More details will be given in Section3on Cartan subalgebras and root systems. 6The definition of κg is not needed. The formula is given for later reference.
the classical Bochner Laplacian acting on C∞(X, F ). By [B11, eq. (2.13.2)], the splitting (2.39) of Cg descends to the splitting of Cg,X,
(2.51) Cg,X = −∆X + Ck,F.
2.5. Orbital integrals and the Casimir. Let µ ∈ Seven(R), and let b
µ ∈ Seven(R) be its Fourier transform, i.e.,
(2.52) µ (y) =b
Z
R
e−2iπyxµ (x) dx.
We assume that there exists C > 0 such that for any k ∈ N, there is ck> 0 such that
(2.53)
b
µ(k)(y) ≤ ckexp −Cy2 .
The above condition is verified if µ has compact support. For t > 0, itb is also verified by the Gaussian function e−tx2.
If A ∈ R, the operator µ√Cg,X+ Ais self-adjoint with a smooth
kernel µ√Cg,X+ A(x, x0) with respect to the Riemannian volume
dx0 on X. As explained in [B11, Section 6.2], using finite propagation speed for the wave equation, condition (2.53) implies that there exist C > 0, c > 0 such that if x, x0 ∈ X, then
(2.54) µ p Cg,X + A(x, x0) ≤ Ce −cd2(x,x0) . If bµ has compact support, then µ
√
Cg,X + A(x, x0) vanishes when
d (x, x0) is large enough.
As explained in [B11, Section 6.2], the above condition guarantees that if γ ∈ G is semisimple, the orbital integral Tr[γ]hµ√Cg,X + Ai
is well-defined. Let us give more details on our conventions.
Let γ ∈ G be taken as in (2.19). Let NX(γ)/X be the orthogonal
bundle to T X (γ) in T X. By [B11, eq. (3.4.1)], we have the identity (2.55) NX(γ)/X = Z0(γ) ×K0(γ)p⊥(γ) .
Let NX(γ)/X be the total space of NX(γ)/X. By [B11, Theorem 3.4.1],
the normal geodesic coordinate system based at X (γ) gives a smooth identification of NX(γ)/X with X. Let dx, dy, df be the Riemannian
volume forms on X, X (γ) , NX(γ)/X. Then dydf is a volume form on
NX(γ)/X. Let r (f ) denote the corresponding Jacobian, so that we have
the identity of volume forms on X,
By [B11, eq. (3.4.36)], there are constants C > 0, C0 > 0 such that
(2.57) r (f ) ≤ CeC0|f |.
By [B11, Theorem 3.4.1], there exists Cγ > 0 such that for f ∈
p⊥(γ) , |f | ≥ 1,
(2.58) dγ efx0 ≥ |a| + Cγ|f | .
As explained in [B11, eq. (4.2.6)], by (2.54), (2.58), there exist Cγ >
0, cγ > 0 such that if f ∈ p⊥(γ), then
(2.59) µ p Cg,X + A γ−1efx 0, efx0 ≤ Cγexp −cγ|f | 2 .
We denote by γ∗ the action of γ on F . More precisely, if x ∈ X, γ∗
maps Fx into Fγx.
In [B11, Definition 4.2.2], the orbital integral Tr[γ]hµpCg,X + Ai
is defined by the formula (2.60) Tr[γ]hµpCg,X + Ai = Z p⊥(γ) Trhγ∗µ p Cg,X + A γ−1efx 0, efx0 i r (f ) df. Equations (2.57), (2.59) guarantee that the integral in (2.60) converges.
Let dk be the Haar measure on K such that Vol (K) = 1. Then dg = dxdk is a Haar measure on G. Let dy be the Riemannian volume form on X (γ). Let dk00 be the Haar measure on K0(γ) such that
Vol (K0(γ)) = 1. Then dz0 = dydk00 is a Haar measure on Z0(γ). Let dv0 be the volume on Z0(γ) \ G such that dg = dz0dv0.
As explained in [B11, Section 4.2], the smooth kernel µpCg,X + A(x, x0)
lifts to a smooth function on G with values in End (E), denoted µEpCg,X + A(g) ,
and by [B11, eq. (4.2.11)], we have the identity
(2.61) Tr[γ]hµpCg,X + Ai = Z Z0(γ)\G TrEhµpCg,X+ A v0−1 γv0idv0.
This definition of orbital integrals coincides with the definition given by Selberg [Se56, p. 66].
2.6. The function Jγ. We use the assumptions in Subsection2.2 and
the corresponding notation. In particular γ ∈ G is a semisimple ele-ment as in (2.19).
Then Ad (γ) preserves z (a) , z⊥(a). Also Ad (k−1) preserves z⊥a (γ). If Yk
0 ∈ k (γ), ad Y0k preserves z ⊥
a (γ). The splitting (2.28) is preserved
by Ad (k−1) and ad Yk 0. If x ∈ R, put (2.62) A (x) =b x/2 sinh (x/2). If Yk 0 ∈ k (γ), ad Y0k
acts as an antisymmetric endomorphism of p(γ) , k (γ), so that its eigenvalues are either 0, or they come by nonzero conjugate imaginary pairs. If Yk
0 ∈ ik (γ), put 7 b A ad Y0k |p(γ) = h detA ad Yb 0k |p(γ) i1/2 , (2.63) b A ad Y0k |k(γ) = h det b A ad Y0k |k(γ) i1/2 .
The square root in (2.63) is the positive square root of a positive real number.
We follow [B11, Theorem 5.5.1], while slightly changing the notation. Definition 2.5. If Yk 0 ∈ ik (γ), put (2.64) Lγ Y0k = det1 − Adk−1e−Y0k |k⊥ a(γ) det 1 − Ad k−1e−Yk 0 |p⊥ a(γ) . Set (2.65) Mγ Y0k = 1 det (1 − Ad (k−1)) | z⊥ a(γ) Lγ Y0k 1/2 .
The fact that the square root in (2.65) is unambiguously defined is established in [B11, Section 5.5]. Let us explain the details. First we make Yk 0 = 0. Then (2.66) 1 det (1 − Ad (k−1)) | z⊥ a(γ) det (1 − Ad (k−1)) |k⊥ a(γ) det (1 − Ad (k−1)) | p⊥ a(γ) = 1 det (1 − Ad (k−1)) | p⊥ a(γ) 2 .
The conventions in [B11] say that the square root of (2.66) is the ob-vious positive square root, i.e.,
(2.67) Mγ(0) = 1 det (1 − Ad (k−1)) | p⊥ a(γ) . Using analyticity in the variable Yk
0 ∈ ik (γ), the choice of the square
root in (2.65) determines a choice of the square root in (2.68). This point will be discussed at length in Section 4. No choice of a Cartan subalgebra or of a positive root system is needed at this stage.
Definition 2.6. Let Jγ Y0k be the smooth function of Y0k ∈ ik (γ),
(2.68) Jγ Y0k = 1 det (1 − Ad (γ)) |z⊥(a) 1/2 b A ad Yk 0 |p(γ) b A ad Yk 0 |k(γ) Mγ Y k 0 .
With the conventions in [B11, Chapter 5], where instead a function Jγ Y0k is defined on k (γ), we have (2.69) Jγ Y0k = Jγ iY0k . By [B11, eq. (5.5.11)] or by (2.68), if Yk 0 ∈ ik, then (2.70) J1 Y0k = b A ad Yk 0 |p b A ad Yk 0 |k .
2.7. Some properties of the function Jγ. Let ρE
∗
: K → U (E∗) denote the representation of K which is dual to the representation ρE. Proposition 2.7. If Yk 0 ∈ ik (γ), then Jγ−1 Y0k = Jγ −Y0k , TrE ∗h ρE∗ke−Y0k i = TrEhρEk−1eY0k i , (2.71) Jγ Y0k = Jγ −Y0k , Tr EρE k−1e−Yk 0 = TrE ∗h ρE∗k−1eY0k i . Proof. If f ∈ End (g), let ef ∈ End (g) denote the adjoint of f with respect to B. We have the identity
(2.72) Ad γ−1 = ^Ad (γ).
By (2.72), we deduce that
A similar argument shows that det1 − Adke−Y0k |p⊥ a(γ) = det 1 − Adk−1eY0k |p⊥ a(γ), (2.74) det1 − Adke−Y0k |k⊥ a(γ) = det 1 − Adk−1eY0k |k⊥ a(γ).
By (2.64), (2.65), (2.68), (2.73), and (2.74), we get the first identity in (2.71). The second identity in (2.71) is trivial.
If Y0k∈ ik (γ), det 1 − Ad k−1e−Yk 0 | p⊥ a(γ) = det 1 − Adk−1eY0k |p⊥ a(γ), (2.75) det 1 − Ad k−1e−Yk 0 | k⊥ a(γ) = det 1 − Adk−1eY0k |k⊥ a(γ).
By (2.75), we get the third identity in (2.71). Since Yk
0 ∈ ik (γ), the
fourth identity is trivial. The proof of our proposition is completed. 2.8. A geometric formula for the orbital integrals associated with the Casimir. Note that ik (γ) is naturally an Euclidean vector space. If Y0k ∈ ik (γ), we denote by Y0k its Euclidean norm. More precisely, if Yk
0 ∈ ik (γ), then
(2.76) Y0k
2
= B Y0k, Y0k .
By [B11, eq. (6.1.1)], there exist c > 0, C > 0 such that if Yk
0 ∈ ik (γ), (2.77) Jγ Y0k ≤ c exp C Y0k . In the sequel,R
ik(γ) denotes integration on the real vector space ik (γ).
Let dY0k be the Euclidean volume on ik (γ). Set p = dim p (γ) , q = dim k (γ). Now we state the result obtained in [B11, Theorem 6.1.1]. Our reformulation takes equation (2.48) into account.
Theorem 2.8. For t > 0, the following identity holds: (2.78) Tr[γ]exp −tCg,X/2 = exp (−tB∗ (ρg, ρg) /2)exp − |a| 2 /2t (2πt)p/2 Z ik(γ) Jγ Y0k Tr EhρEk−1 e−Y0k i exp− Y0k 2 /2t dY k 0 (2πt)q/2. Let B|z(γ) be the restriction of B to z (γ), and let B∗|z(γ) be the
corresponding quadratic form on z∗(γ). Let ∆z(γ)denote the associated
generalized Laplacian on z (γ). We can extend ∆z(γ) to an operator
Put
(2.79) zi(γ) = p (γ) ⊕ ik (γ) .
Then B|zi(γ) is a scalar product on zi(γ). The generalized Laplacian
∆z(γ) restricts on z
i(γ) to the standard Euclidean Laplacian of zi(γ).
We take µ ∈ Seven(R) as in Subsection 2.5. If f ∈ z
i(γ), let µ q −∆z(γ)+ B∗(ρg, ρg) + A (f )
be the smooth convolution kernel for µp−∆z(γ)+ B∗(ρg, ρg) + A
on zi(γ) with respect to the volume form associated with the scalar
product of zi(γ). Using (2.53) and finite propagation speed for the
wave equation, there exist C > 0, c > 0 such that if f ∈ zi(γ), then
(2.80) µ q −∆z(γ)+ B∗(ρg, ρg) + A (f ) ≤ Ce−c|f |2.
Let δa be the Dirac mass at a ∈ p (γ). Then
Jγ Y0k Tr
EhρEk−1
e−Y0k
i δa
is a distribution on zi(γ), to which the smooth convolution kernel
µ q
−∆z(γ)+ B∗(ρg, ρg) + A
can be applied. By definition, (2.81) µ q −∆z(γ)+ B∗(ρg, ρg) + A h Jγ Y0k Tr EhρEk−1 e−Y0k i δa i (0) = Z ik(γ) µ q −∆z(γ)+ B∗(ρg, ρg) + A −Yk 0, −a Jγ Y0k TrE h ρEk−1e−Y0k i dY0k.
In the right hand-side of (2.81), −Y0k, −a can as well be replaced by Y0k, a.
In [B11, Theorem 6.2.2], the following extension of Theorem2.8 was established.
Theorem 2.9. The following identity holds: (2.82) Tr[γ]hµpCg,X + Ai= µ q −∆z(γ)+ B∗(ρg, ρg) + A h Jγ Y0k Tr EhρEk−1 e−Y0k i δa i (0) .
3. Cartan subalgebras, Cartan subgroups, and root systems
The purpose of this Section is to recall basic facts on Cartan subal-gebras, on Cartan subgroups, and on root systems.
This section is organized as follows. In Subsection3.1, we state some elementary facts of linear algebra.
In Subsection 3.2, we recall the definition of Cartan subalgebras. In Subsection 3.3, we introduce the corresponding root system, and the associated algebraic Weyl group.
In Subsection 3.4, we define the real and the imaginary roots. In Subsection 3.5, we construct a positive root system.
In Subsection 3.6, when the Cartan subalgebra is fundamental, we compare the root system of k with the root system of g.
In Subsection 3.7, we introduce the Cartan subgroups, and of the corresponding regular elements.
In Subsection 3.8, we relate semisimple elements in G to Cartan subgroups.
In Subsection 3.9, we describe the characters of Cartan subgroups associated with a root system.
In Subsection3.10, we give some properties of the real and imaginary roots with respect to semisimple elements in G.
Finally, in Subsection3.11, we give a well-known formula that relates the action of invariant differential operators on the Lie algebra g and on a Cartan subalgebra h.
We make the same assumptions as in Section 2, and we use the corresponding notation.
3.1. Linear algebra. Let V be a finite dimensional real vector space. The symmetric algebras S·(V ) , S·(V∗) are the algebras of polynomials on V∗, V . If v ∈ V , v acts as a derivation of S·(V∗). More generally, S·(V ) acts on S·(V∗), and this action identifies S·(V ) with the algebra D·(V ) of real partial differential operators on V with constant coeffi-cients. In particular, if B∗ ∈ S2(V ) is a bilinear symmetric form on V∗,
B∗ is positive, ∆V is just a classical Laplacian. If B∗ is negative, then ∆V is the negative of a classical Laplacian on V .
Let VC = V ⊗R C be the complexification of V , a complex
vec-tor space. Its complex dual is given by VC∗ = V∗ ⊗R C. The
alge-bras S·(VC) , S·(VC∗) are the algebras of complex polynomials on V ∗, V . Note that S·(VC) = S·(V ) ⊗RC, S·(VC∗) = S · (V∗) ⊗RC. (3.1) Put D·(VC) = D (V ) ⊗RC, D·(VC∗) = D · (V∗) ⊗RC. (3.2)
Then D·(VC) , D·(VC∗) are the complexifications of D
·(V ) , D·(V∗),
and also the spaces of complex holomorphic differential operators with constant coefficients on VC, VC∗.
In particular, if B∗ ∈ S2(V ), ∆V is now viewed as a holomorphic
operator on VC, that coincides with the corresponding Laplacian ∆V
on V , and with −∆V on iV ' V . 8
Also S·(V∗) ⊂ C∞(V, R), and the action of D·(V ) extends to C∞(V, R).
Let S [[V∗]] be the algebra of formal power series s = P+∞ i=0s
i, si ∈
Si(V∗). Then S [[V∗]] can be identified with the algebra D [[V∗]] of
differential operators of infinite order with constant coefficients on V∗. In particular, S [[V∗]] acts on S·(V ).
3.2. The Cartan subalgebras of g. By [W88, Section 0.2], a Lie subalgebra h ⊂ g is said to be a Cartan subalgebra if h is maximal among the abelian subalgebras of g whose elements act as semisim-ple endomorphisms of g. Cartan subalgebras are known to exist and have the same dimension r, which is called the complex rank of G. By [Kn02, Proposition 6.64], there is a finite family of nonconjugate Cartan subalgebras in g. By [W88, Lemma 2.3.3], in every conjugacy class of Cartan subalgebras, there is a unique θ-stable Cartan algebra, up to conjugation by K. Therefore there is a finite family of non-conjugate θ-stable Cartan subalgebras, up to conjugation by K. By [Kn02, Theorem 2.15], the Cartan subalgebras of gC are unique up to
automorphisms induced by the adjoint group Ad (gC).
8On C ' R2, when acting on holomorphic functions, the differential operators ∂
∂z, ∂ ∂x, −i
∂
∂y coincide. In this sense, the differential operator ∂
∂xon R extends to the
differential operator ∂
∂z on C, and restricts to the operator −i ∂
∂y on the imaginary
line iR. The operator ∂x∂22 on R restricts to the operator −
∂2
Let h ⊂ g be a θ-stable Cartan subalgebra. To the Cartan splitting of g in (2.1) corresponds the splitting
(3.3) h= hp⊕ hk.
In particular the restriction B|h of B to h is nondegenerate. This is
also the case if h is any Cartan subalgebra.
Up to conjugation by K, there is a unique θ-stable Cartan subalgebra h, which is called fundamental, such that hk is a Cartan subalgebra of
k. Let us give more details on its construction [Kn86, pp. 129, 131]. Let t ⊂ k be a Cartan subalgebra of k. Let z (t) ⊂ g be the centralizer of t, i.e.,
(3.4) z(t) = {f ∈ g, [t, f ] = 0} .
Then h = z (t) is a θ-stable fundamental Cartan subalgebra of g, and hk = t.
An element f ∈ g is said to be semisimple if ad (f ) ∈ End (g) is semisimple. If h is a Cartan subalgebra, elements of h are semisimple. Any semisimple element of g lies in a Cartan subalgebra.
If h ⊂ g is a Cartan subalgebra, let h⊥ be the orthogonal to h in g. We have the B-orthogonal splitting,
(3.5) g= h ⊕ h⊥,
and B is also nondegenerate on h⊥. If h is θ-stable, then h⊥ is also θ-stable, and so it splits as
(3.6) h⊥ = h⊥p ⊕ h⊥k.
Let u = ip ⊕ k be the compact form of g. Then hu = ihp⊕ hk is a
Cartan subalgebra of u. If h is θ-stable, then hu is also θ-stable.
An element f ∈ g is said to be regular if z (f ) is a Cartan subalgebra. Regular elements in g are semisimple.
If h is a Cartan subalgebra, if f ∈ h, ad (f ) acts as an endomorphism of g/h. Then f ∈ h is regular if and only if det ad (f ) |g/h6= 0.
3.3. A root system and the Weyl group. Let h be a θ-stable Car-tan subalgebra.
Let R ⊂ h∗C be the root system associated with h, g [Kn02, Section II.4]. If α ∈ R, then −α ∈ R, α ∈ R. If α ∈ R, let gα ⊂ gC be the
weight space associated with α, which is of dimension 1. Then we have the splitting (3.7) gC = hC M ⊕α∈Rgα. If α ∈ R, then (3.8) gα = gα.
If f ∈ h, ad (f ) ∈ End (g) is antisymmetric with respect to B, so that the gα|α∈R are B-orthogonal to hC. If α, β ∈ R, then gα, gβ are
B-orthogonal except when β = −α, and the pairing between gα, g−α is
nondegenerate, so that if α ∈ R, the form B induces the identification
(3.9) g−α ' g∗α.
Also
(3.10) h⊥C = ⊕α∈Rgα.
If α ∈ R, α takes real values on hp, and imaginary values on hk, i.e.,
α ∈ h∗p ⊕ ih∗
k. Also θ preserves the splitting (3.5) of g, and it maps R
into itself. More precisely, if α ∈ R,
θα = −α, gθα = θgα.
(3.11)
Let W (hC: gC) ⊂ Aut (hC) be the algebraic Weyl group [Kn86, p.
131]. Then R ⊂ ih∗u, and W (hC : gC) ⊂ Aut (hu), i.e., W (hC: gC)
preserves the real vector space hu. Also θ acts as an automorphism of
the Lie algebras g, u, and W (hC, gC) is preserved by conjugation by θ.
In general, W (hC : gC) does not preserve the real vector space h.
Recall that if h ∈ hC, we can define its complex conjugate h ∈ hC.
If u ∈ End (hC), its complex conjugate u ∈ End (hC) is such that if
h ∈ hC, then
(3.12) u (h) = u h.
Proposition 3.1. If w ∈ W (hC : gC), then
(3.13) w = θwθ−1.
In particular, the group W (hC: gC) is preserved by complex
conjuga-tion.
Proof. The group W (hC: gC) is generated by the symmetries sα, α ∈ R
with respect to the vanishing locus of the α ∈ R. By (3.11), we deduce that if α ∈ R,
(3.14) sα = θsαθ−1,
from which get (3.13). Since W (hC : gC) is stable by conjugation by
θ, the group W (hC: gC) is preserved by complex conjugation.
Another proof is as follows. Observe that there is a canonical iden-tification of complex vector spaces ϕ : hC ' hu,C, but the complex
conjugations on hC and on hu,C are not the same. More precisely, if
h ∈ hC,
If w ∈ W (hC: gC), then
(3.16) w|hC = ϕ
−1
w|hu,Cϕ.
Recall that w is a real automorphism of the real vector space hu. By
(3.15), (3.16), we get
(3.17) w|hC = ϕ
−1w| hu,Cϕ.
By (3.15)–(3.17), we get (3.13). The proof of our proposition is
com-pleted.
3.4. Real roots and imaginary roots. Let Rre ⊂ R be the roots
α ∈ R such that θα = −α, let Rim be the roots α ∈ R such that
θα = α. These are respectively the real roots and the imaginary roots. Imaginary roots vanish on hp, real roots vanish on hk. By [Kn86, p.
349], the set of complex roots Rc ⊂ R is defined to be
(3.18) Rc= R \ Rre∪ Rim .
Proposition 3.2. If α ∈ R, the map f ∈ gC → f ∈ gC induces an
antilinear isomorphism from gαinto g−θα, and the map f ∈ gC→ θf ∈
gC induces an antilinear isomorphism from gα into g−α. If α ∈ Rre,
gα is the complexification of a real vector subspace of h⊥.
Proof. If b ∈ h, f ∈ gα, then
(3.19) [b, f ] = hα, bi f.
By taking the conjugate of (3.19), we obtain
(3.20) b, f = h−θα, bi f .
By (3.20), we get the first part of our proposition. By composing this isomorphism with θ, we obtain the second part. If α ∈ Rre, then −θα = α, so that gα is real. The proof of our proposition is completed.
Definition 3.3. Put
i= ker ad (hp) ∩ h⊥, r= ker ad (hk) ∩ h⊥.
(3.21)
Then θ acts on i, r, so that we have the splittings,
i= ip⊕ ik, r= rp⊕ rk.
(3.22)
Proposition 3.4. The vector spaces i, r are orthogonal in h⊥. More-over,
iC= ⊕α∈Rimgα, rC = ⊕α∈Rregα.
(3.23)
If α ∈ Rim, then either g
Proof. If f ∈ h⊥, f ∈ i∩ r, then f commutes with h. Since h is a Cartan subalgebra, f = 0, so that i ∩ r = 0. If α ∈ R \ Rim, α does not vanish
identically on hp, and its vanishing locus in hp is a hyperplane. So one
can find bp∈ hp\ 0 such that for any α ∈ R \ Rim, hα, bpi 6= 0. Then
(3.24) i= ker ad (bp) ∩ h⊥.
Since i∩r = 0, ad (bp) acts as an invertible morphism of r. Therefore any
element of r lies in the image of ad (bp). Since ad (bp) is symmetric in the
classical sense, i and r are orthogonal. Equation (3.23) is elementary. If α ∈ Rim, the action of h on g
α factors through hk. Also ad (hk)
preserves the splitting g = p ⊕ k. Therefore if α ∈ Rim, either g
α ⊂ pC,
or gα⊂ kC. The proof of our proposition is completed.
Definition 3.5. Put Rimp =α ∈ Rim, g α ⊂ pC , Rimk =α ∈ R im, g α ⊂ kC . (3.25) Then (3.26) Rim = Rimp ∪ Rim k .
Let c denote the orthogonal to i ⊕ r in h⊥. Again c splits as
(3.27) c= cp⊕ ck.
Moreover, we have the orthogonal splitting
(3.28) h⊥= i ⊕ r ⊕ c.
Proposition 3.6. The following identity holds:
(3.29) cC = ⊕α∈Rcgα.
Proof. This follows from equations (3.10), (3.23), and (3.28). Now we give a result taken from [W88, Lemma 2.3.5].
Proposition 3.7. A θ-stable Cartan subalgebra h is fundamental if and only if there are no real roots.
Proof. If α ∈ R, then α ∈ Rre if and only if when f ∈ gα,
(3.30) [hk, f ] = 0.
If h is fundamental, by (3.4), then f ∈ hC, so that f = 0, which
proves there are no real roots. Conversely, if there is α ∈ Rre, then gα ⊂ z (hk)C, so that z (hk) is not equal to h, and h is not fundamental.
Proposition 3.8. The vector spaces ip, ik, cp, ck have even dimension.
Also H ∩ K preserves these vector spaces, and the corresponding deter-minants are equal to 1. Also rp, rk (resp. cp, ck) have the same
dimen-sion, and the actions of H ∩K on these two vector spaces are equivalent. In particular, we have the identity
(3.31) dim p − dim hp = dim ip+
1
2dim r + 1 2dim c.
Proof. If α ∈ R \ Rre, the vanishing locus of α in hk is a hyperplane.
Therefore we can find fk ∈ hk\ 0 such for any α ∈ R \ Rre, hα, fki 6= 0,
which just says that ad (fk) acts as an invertible endomorphism of i, c.
This endomorphism preserves their p and k components, and it is clas-sically antisymmetric. This is only possible if these vector spaces are even dimensional. If k ∈ H ∩ K, Ad (k−1) preserves these vector spaces and commutes with ad (fk). Therefore the eigenspaces associated with
the eigenvalue −1 are preserved by ad (fk), so they are even
dimen-sional. This forces the determinant of Ad (k−1) to be equal to 1 on each of these vector spaces.
We choose bp ∈ hp\ 0 such that for any α ∈ R \ Rim, hα, bpi 6= 0.
Therefore ad (bp) acts as an automorphism of r, c that exchanges the
corresponding p and k parts, and commutes with Ad (k−1). By (3.28), we get
(3.32) h⊥p = ip⊕ rp⊕ cp.
Using the results we already established and (3.32), we get (3.31). The
proof of our proposition is completed.
3.5. A positive root system. Let h be a θ-stable Cartan subalgebra, and let R denote the corresponding root system. Let R+ ⊂ R be a
positive root system. Set
R+re= R+∩ Rre, Rim+ = R+∩ Rim, Rc+ = R+∩ Rc, (3.33) so that (3.34) R+= Rre+ ∪ R im + ∪ R c +.
In the whole paper, we choose R+ such that −θ preserves R+\ Rim+.
Equivalently, we assume that if α ∈ R+\ Rim+, then α ∈ R+.
Let us explain how to do this. If α ∈ R \ Rim, the vanishing locus of α in hp is a hyperplane, and so there is bp ∈ hp, |bp| = 1 such that
for any α ∈ R \ Rim, hα, b
pi 6= 0. The same argument shows that there
is bk ∈ hk, |bk| = 1 such that for α ∈ Rim, hα, bki 6= 0. For > 0,
small enough, for α ∈ R, the real numbers hα, b±i do not vanish, and
if α ∈ R \ Rim, they have opposite signs. Put
(3.35) R+= {α ∈ R, hα, b+i > 0} .
Then R+ is a positive root system such that −θ preserves R+\ R+im.
Note that −θ acts without fixed points on Rc+, so that Rc+ is even. Definition 3.9. Put
c+,C= ⊕α∈Rc
+gα, c−,C = ⊕α∈−Rc+gα.
(3.36)
Proposition 3.10. The vector spaces c+,C, c−,C are the
complexifica-tions of real Lie subalgebras c+, c− of g, which have the same even
dimension, and are such that
c= c+⊕ c−, c− = θc+.
(3.37)
Also B vanishes on c+, c− and induces the identification,
(3.38) c− ' c∗+.
The projections on p, k map c± into cp, ck isomorphically. Finally, the
actions of H ∩ K on c+, c−, cp, ck are equivalent.
Proof. By Proposition 3.2, c+,C, c−,C are stable by conjugation, and
so they are complexifications of real vector spaces c+, c−. The fact
that these are Lie subalgebras is obvious. Since Rc+ is even, these vector spaces are even dimensional, and also they have the same di-mension. By Proposition 3.2, θ induces an isomorphism of c+ into c−.
Using the considerations that follow (3.9), we find that B vanishes on c+, c−, and we obtain (3.38). By Proposition 3.8, c+, c−, cp, ck have the
same even dimension. The projections on p, k are given respectively by
1
2(1 ∓ θ). Since θ exchanges c+ and c−, they restrict to isomorphisms
on c+, c−. By Proposition3.8, we know that the actions H ∩ K on cp, ck
are equivalent. Since the adjoint action of H ∩ K commutes with θ, the corresponding representations of H ∩ K on these vector spaces are
equivalent. The proof of our proposition is completed.
If f ∈ h, then (3.39) det ad (f )h⊥ = Y α∈R hα, f i . Definition 3.11. Let πh,g ∈ S·(h∗
C) be such that if h ∈ hC, then
(3.40) πh,g(h) = Y
α∈R+
By (3.39), if f ∈ h,
(3.41) det ad (f )|
h⊥
= πh,g(f ) πh,g(−f ) .
Also f ∈ h is regular if and only if πh,g(f ) 6= 0.
Proposition 3.12. The function πh,g vanishes identically on h
k if and
only if h is not fundamental.
Proof. Assume that h is not fundamental. By Proposition 3.7, there are real roots, and so there are real positive roots. If α ∈ Rre+, then α vanishes on hk, and so πh,g vanishes on hk. Conversely, if πh,g vanishes
on hk, one of the α ∈ R+ has to vanish identically on hk, so that
α ∈ Rre+, and h is not fundamental. The proof of our proposition is
completed.
3.6. The case when h is fundamental and the root system of (hk, k). In this Subsection, we assume that h is a θ-stable fundamental
Cartan subalgebra of g. By (3.18), we get
(3.42) Rc= R \ Rim.
By Proposition3.4, r = 0. By (3.28), we have the orthogonal splitting, h⊥p = ip⊕ cp, h⊥k = ik⊕ ck.
(3.43)
Also ip, ik have even dimension, cp, ck have the same even dimension,
and the action of H ∩ K on these last two vector spaces are conjugate. The roots in R do not vanish identically on hk. We will now reinforce
the choice of positive roots made in Subsection 3.5. We may and we will assume that bk ∈ hk, |bk| = 1 has been chosen so that if α ∈ R,
hα, bki 6= 0.
As we saw in Subsection 3.5, −θ acts without fixed points on R+c. Also if α ∈ Rc+, −θα|hk = −α|hk, so that the nonzero real numbers
hα, ibki and h−θα, ibki have opposite signs.
Set
Rimk,+ = Rimk ∩ R+, Rc++ =α ∈ R c
+, hα, ibki > 0 .
(3.44)
Definition 3.13. Let R (hk, k) be the root system associated with the
pair (hk, k). If R+(hk, k) is a positive root system for (hk, k), if hk∈ hk,C,
put (3.45) πhk,k(h k) = Y β∈R+(hk,k) hβ, hki .
Then πhk,k2(h
k) does not depend on the choice of R+(hk, k). The
arguments above (3.44) show that if h ∈ hk, then
(3.46) Y
α∈Rc +
hα, hki ≥ 0.
Proposition 3.14. The map α ∈ Rim
k ∪ R+c → α|hk is injective, and
gives the identification
(3.47) R (hk, k) = Rkim∪ R
c +.
A positive root system R+(hk, k) for (hk, k) is given by
(3.48) R+(hk, k) = Rimk,+∪ R c ++. If hk∈ hk,C, then (3.49) πhk,k(h k) 2 = (−1)12|R c +| Y α∈Rim k,+ hα, hki 2 Y α∈Rc + hα, hki . Proof. By (3.28), we get (3.50) h⊥k = ik⊕ ck,
and the above splitting is preserved by hk. The weights for this action
on ik are given by Rimk . By Proposition 3.10, ck and c+ are equivalent
under the action of hk. By the first equation in (3.36), the weights for
the action of hk on c+ are given by the restriction of R+c to hk. Since
the weights for the action of hk on ck are nonzero and of multiplicity
1, the map α ∈ Rimk ∪ Rc
+ → α|hk gives the identification in (3.47). By
(3.47), we get (3.48). Using (3.45) and the above results, we get (3.49).
The proof of our proposition is completed.
Remark 3.15. The results contained in Proposition 3.14 will play an important role in the proof of the limit results of Subsection 8.1. 3.7. Cartan subgroups and regular elements. If h is a Cartan subalgebra, the associated Cartan subgroup H ⊂ G is the stabilizer of h. Then H is a Lie subgroup of G, with Lie algebra h.
Assume that h is θ-stable. Then θ restricts to an involution of H, and (3.3) is the corresponding Cartan splitting of h. Also B restricts to a H and θ invariant symmetric nondegenerate bilinear form B|h on
h, so that H is a reductive subgroup of G.
We still assume h to be θ-stable. Let ZG(H) ⊂ G be the centralizer
in H, it is just the center Z (H) of H. As in [Kn86, p. 131], the analytic Weyl group W (H : G) is defined as the quotient
(3.51) W (H : G) = NG(H) /ZG(H) .
Put
ZK(H) = ZG(H) ∩ K, NK(H) = NG(H) ∩ K.
(3.52)
Then NK(H) /ZK(H) embeds in W (H : K). By [Kn86, p. 131], this
embedding is an isomorphism, i.e.,
(3.53) W (H : G) = NK(H) /ZK(H) .
By [Kn86, eq. (5.6)], W (H : G) ⊂ W (hC: gC).
By [Kn86, p. 130], an element γ ∈ G is said to be regular if z (γ) is a Cartan subalgebra. If H is the corresponding Cartan subgroup, then γ ∈ H. By [Kn86, Theorem 5.22], the set Greg ⊂ G of regular elements is open and conjugation invariant. More precisely, if H1, . . . , H`denotes
the finite family of nonconjugate Cartan subgroups, by [Kn86, Theorem 5.22], Greg splits as the disjoint union of open sets
(3.54) Greg = ∪`i=1GregH
i,
where GregH
i denote the open set of elements of G
reg that are conjugate
to an element of Hi.
If γ ∈ H, Ad (γ) acts on g and fixes h. Since Ad (γ) preserves B, it also acts on h⊥, so that 1 − Ad (γ) acts on h⊥. Then γ is regular if and only this endomorphism is invertible, i.e., det (1 − Ad (γ)) |h⊥ 6= 0.
3.8. Cartan subgroups and semisimple elements. The following result is established in [Va77, Part I, Section 2.3, Theorem 4].
Proposition 3.16. A group element γ ∈ G is semisimple if and only if it lies in a Cartan subgroup.
Let us give a direct proof of part of our proposition. Le h be a θ-stable Cartan subalgebra, and let H be the corresponding Cartan subgroup. If γ ∈ H, then h ⊂ z (γ). Moreover, γ can be written uniquely in the form
γ = eak−1, a ∈ hp, k ∈ H ∩ K.
(3.55)
Since a ∈ hp, k ∈ H, then Ad (k) a = a, which guarantees that γ is
semisimple in G.
Let h ⊂ g be a Cartan subalgebra, and let H ⊂ G be the associated Cartan subgroup. If γ ∈ H, then h ⊂ z (γ), so that h is a Cartan subalgebra of z (γ). In particular G and Z0(γ) have the same complex
Proposition 3.17. Any θ-stable Cartan subalgebra h0 of z (γ) is also
a Cartan subalgebra of g.
Proof. As we saw in Subsection 2.2, Z0(γ) is a connected reductive
group and θ induces on Z0(γ) a corresponding Cartan involution. Since
h0 is commutative and θ-stable, and since its action on g preserves B,
it acts on g by semisimple endomorphisms of g. Since G and Z0(γ)
have the same complex rank, h0 is a Cartan subalgebra of g. The proof
of our proposition is completed.
3.9. Root systems and their characters. Let h ⊂ g be a θ-stable Cartan subalgebra. We use the notation of the previous subsections.
Take γ ∈ H. As we saw after Proposition 3.16, if γ ∈ H, we can write γ uniquely in the form
γ = eak−1, a ∈ hp, k ∈ H ∩ K,
(3.56) so that
(3.57) Ad (k) a = a.
Let R (γ) , R (a) be the root systems associated with (h, z (γ)) , (h, z (a)). We will denote with extra subscripts the corresponding real, imaginary, and complex roots.
Theorem 3.18. If γ ∈ H, for any α ∈ R, Ad (γ) preserves the 1-dimensional complex line gα. For every α ∈ R, there is a character
ξα : H → C∗ such that Ad (γ) acts on gα by multiplication by ξα(γ).
If α ∈ R,
(3.58) ξαξ−α = 1.
If α ∈ R, if f ∈ h, k ∈ H ∩ K, then
ξα ef = ehα,f i, |ξα(k)| = 1.
(3.59)
In particular, if γ ∈ H is taken as in (3.56), then
ξα(γ) = ehα,aiξα k−1 , ξ−θα(γ) = ξα(γ).
(3.60)
If α ∈ Rre, then ξ
α(γ) ∈ R∗, if α ∈ Rim, then |ξα(γ)| = 1. If α ∈ Rre,
the restriction of ξα to H ∩ K takes its values in {−1, +1}.
Also
(3.61) det (1 − Ad (γ)) |h⊥ =
Y
α∈R
(1 − ξα(γ)) ,
The following identities hold:
R (γ) = {α ∈ R, ξα(γ) = 1} , R (a) = {α ∈ R, hα, ai = 0} ,
Rre(γ) = R (γ) ∩ Rre, Rim(γ) = R (γ) ∩ Rim, Rc(γ) = R (γ) ∩ Rc, (3.62)
Rre(a) = R (a) ∩ Rre, Rim(a) = R (a) ∩ Rim, Rc(a) = R (a) ∩ Rc. Also R+(γ) = R (γ)∩R+, R+(a) = R (a)∩R+ are positive root systems
for (h, z (γ)) , (h, z (a)).
Proof. If γ ∈ H, then Ad (γ) fixes h, and so if h ∈ h, we have the commutation relation in End (g),
(3.63) [Ad (γ) , ad (h)] = 0.
By (3.7), (3.63), we deduce that for any α ∈ R, Ad (γ) preserves gα.
Since gα is a complex line, H acts on gα via a character ξα.
Since Ad (γ) preserves B, if f, f0 ∈ gC, we get
(3.64) B (Ad (γ) f, f0) = B f, Ad (γ)−1f0 . Take α ∈ R. By (3.64), if f ∈ gα, f0 ∈ g−α, then
(3.65) ξα(γ) B (f, f0) = ξ−1−α(γ) B (f, f 0
) .
As we saw in Subsection 3.3, if α ∈ R, the pairing between gα and g−α
via B is nondegenerate. By (3.65), we get (3.58).
The first equation in (3.59) is trivial. Since ξαrestricts to a character
of the compact group H ∩ K, we get the second equation in (3.59). The first equation in (3.60) follows from the previous considerations. Since θ (γ) = e−ak−1, and since θ maps gα into gθα, we obtain the
second equation in (3.60). From this second equation, we deduce that if α ∈ Rre, then ξ
α(γ) is real, and if α ∈ Rim, then |ξα(γ)| = 1. If
γ ∈ H ∩ K, α ∈ Rre, we know that ξ
α(γ) ∈ R∗, |ξα(γ)| = 1, so that
ξα(γ) = ±1.
Equations (3.61), (3.62) are trivial. By (3.61), γ is regular if and only if for α ∈ R, ξα(γ) 6= 1.
Now we proceed as in [BL99, Theorem 1.38]. If κ ⊂ h is a positive Weyl chamber for (h, g), the forms in R do not vanish on κ, so that κ is included in a z (γ) Weyl chamber. It follows that R (γ) ∩ R+ is
a positive root system on (h, z (γ)). The same argument is valid for
R (a). The proof of our theorem is completed.
3.10. Real roots, imaginary roots, and semisimple elements. We still take γ as in Subsection 3.9. When taking the intersection of i, r, c with z (γ) , z (a) , z (k), this will be indicated with a paren-thesis containing the corresponding argument. The intersection with
z⊥(γ) , z⊥(a) , z⊥(k) will be denoted with an extra ⊥. These vector spaces also have a p and a k component.
By construction,
(3.66) Rim(γ) = Rim(k) .
As in (3.25), (3.26), we get
Rim(γ) = Rimp (γ) ∪ Rimk (γ) , Rim(k) = Rimp (k) ∪ Rkim(k) . (3.67)
To make the notation simpler, in (3.67), we did not use instead the notation p (γ) , k (γ) , p (k) , k (k).
Proposition 3.19. The following identities holds:
i⊂ z (a) , i(γ) = i (k) , i⊥(k) ⊂ z⊥a (γ) . (3.68) Also i(k)C= ⊕α∈Rim(k)gα, i⊥(k) C= ⊕α∈Rim\Rim(k)gα. (3.69) Moreover, z(a)C= hC M ⊕α∈R(a)gα, z⊥(a) ⊂ r ⊕ c, (3.70) z⊥(a)C= ⊕α∈R\R(a)gα. If γ is regular, then (3.71) i(k) = 0.
Proof. By the first identity in (3.21), since a ∈ hp, we get the first
identity in (3.68). Combining the third identity in (2.20) with this first identity, we get the second identity in (3.68). The third identity in (3.68) is a consequence of the first two. By (3.7), we get (3.69) and the first and the third equations in (3.70), the second equation being a consequence of the first equation in (3.68). Also γ is regular if and only if z (γ) = h. Since h ∩ i = 0, by the second identity in (3.68), we
get (3.71). The proof of our proposition is completed.
3.11. Cartan subalgebras and differential operators. Let h be a θ-stable Cartan subalgebra. There is a natural projection g∗ → h∗.
By (3.5), there is a well-defined projection g → h. To the splitting (3.5) corresponds the dual splitting
(3.72) g∗ = h∗ ⊕ h∗⊥.
The projections S·(g∗) → S·(h∗) , S·(g) → S·(h) associated with (3.5), (3.72) are just the restriction r of polynomials on g to h, or of polyno-mials on g∗ to h∗.
Definition 3.20. Let I·(g∗) ⊂ S·(g∗) be the algebra of invariant el-ements in S·(g∗), i.e., the algebra of the elements of S·(g∗) on which the derivations associated with g vanish. Let I·(h∗C, g∗C) be the algebra of W (hC: gC)-invariant elements in S·(h∗C).
Recall that
(3.73) S·(h∗C) = S·(h∗)C.
In particular S·(h∗C) is equipped with a natural conjugation.
Proposition 3.21. The algebra I·(h∗C, g∗C) is preserved under complex conjugation. There is a real algebra I·(h∗, g∗) ⊂ S·(h∗) such that (3.74) I·(h∗C, g∗C) = I·(h∗, g∗)C.
The map r : S·(g∗) → S·(h∗) induces the canonical isomorphism (3.75) r : I·(g∗) ' I·(h∗, g∗) .
Proof. By Proposition 3.1, W (hC: gC) is preserved by conjugation.
Therefore I·(h∗C, g∗C) is preserved by conjugation, which gives (3.74). From the obvious isomorphism
(3.76) r : I·(g∗C) → I·(h∗C, g∗C) ,
we get (3.75). The proof of our proposition is completed. What we did for g∗ can also be done for g. The same argument as in (3.75) leads to the identification
(3.77) r : I·(g) ' I·(h, g) .
As we saw in Subsection3.1, S·(g) can be identified with the algebra D·(g) of real differential operators on g with constant coefficients, so that I·(g) is identified with the algebra D·I(g) of real differential oper-ators with constant coefficients on g which commute with the above g-derivations. Similarly I·(h, g) can be identified with the algebra D·I(h, g) of real differential operators on h with constant coefficients that are W (hC : gC)-invariant.
Let R+ ⊂ R be a positive root system as in Subsection 3.5. Recall
that the associated polynomial πh,g ∈ S·(h∗
C) was defined in (3.40).
If A ∈ I·(g) = DI(g), if f ∈ I·(g∗), then Af ∈ I·(g∗), so that
r (Af ) ∈ I·(h∗, g∗). Also r (A) ∈ I·(h, g) = DI(h, g). By [HC57a,
Lemmas 6 and 8], if f ∈ I·(g∗),
(3.78) r (Af ) = 1
πh,gr (A) π h,grf.
Let C∞,g(g, R) be the vector space of smooth real functions on g that vanish under the above g-derivations. Then (3.78) extends to f ∈ C∞,g(g, R).
4. Root systems and the function Jγ
The purpose of this Section is to give a drastically simplified version of the function Jγ Y0k introduced in Definition2.6. This will be done
by expressing this function in terms of a positive root system. Imagi-nary roots will play an essential role in this expression. In particular, the function Lγintroduced in Definition2.5will turn out not to depend
on a.
This section is organized as follows. In Subsection 4.1, if h is a θ-stable Cartan subalgebra and H is the corresponding Cartan subgroup, if γ ∈ H, we give explicit formulas for the determinant of 1 − Ad (γ) on various subspaces in terms of a positive root system.
In Subsection 4.2, we establish our formula for Jγ Y0k
using the root system.
We use the assumptions and the notation of Section 3.
4.1. The determinant of 1 − Ad (γ). Let h ⊂ g be a θ-stable Cartan subalgebra, and let H ⊂ G be the corresponding Cartan subgroup. Put h⊥+= M α∈R+ gα, h⊥−= M α∈R+ g−α. (4.1) By (3.10), we get (4.2) h⊥C= h⊥+⊕ h⊥−.
Let γ ∈ H be written as in (3.56). By Theorem 3.18, we obtain
(4.3) det Ad (γ) |h⊥
+ =
Y
α∈R+
ξα(γ) .
We write (4.3) in the form (4.4) det Ad (γ) |h⊥ + = Y α∈Rc + ξα(γ) Y α∈Rre + ξα(γ) Y α∈Rim + ξα(γ) .
By the considerations we made in Subsection3.5, −θ acts without fixed points on Rc
+. By Theorem 3.18, in the right-hand side of (4.4), the
first term is positive, the second is a product of nonzero real numbers, and the third term is a product of complex numbers of module 1.
If α ∈ R+, we choose a square root ξ 1/2
α (k−1) of ξα(k−1). In view of
the second identity in (3.60), if α ∈ Rc
+, we may and we will assume
that
For α ∈ R+, we choose the square root ξ 1/2 α (γ) so that (4.6) ξα1/2(γ) = ehα,ai/2ξα1/2 k−1 . By (4.5), (4.6), if α ∈ Rc +, then (4.7) ξ−θα1/2 (γ) = ξα1/2(γ).
A square root of det Ad (γ) |h⊥
+ in (4.3) is given by (4.8) det Ad (γ) |1/2h⊥ + = Y α∈R+ ξα1/2(γ) .
By proceeding as in (4.4), we can rewrite (4.8) in the form (4.9) det Ad (γ) |1/2h⊥ + = Y α∈Rc + ξα1/2(γ) Y α∈Rre + ξα1/2(γ) Y α∈Rim + ξα1/2(γ) .
Using (4.6), (4.7), we find that the first product in the right hand-side of (4.9) is positive, the second product is either a nonzero real number, or the product of√−1 by a nonzero real number, and the third product is of module 1. Definition 4.1. Put (4.10) D(γ) = sgn Y α∈Rre +\Rre+(γ) 1 − ξα−1(γ) .
Recall that if α ∈ Rre, then ξ
α(k−1) = ±1.
Proposition 4.2. The following identity holds:
(4.11) D(γ) = sgn Y α∈Rre +\Rre+(a) 1 − ξ−1α (γ) . Proof. If α ∈ R (a), by (3.60), ξα(γ) = ξα(k−1). If α /∈ R (γ), by
(3.62), ξα(γ) 6= 1. By Theorem 3.18, if α ∈ Rre(a) \ Rre(γ), we have
ξα(γ) = −1, so that 1 − ξα−1(γ) = 2. This completes the proof of our
Theorem 4.3. The following identities hold: det (1 − Ad (γ)) |z⊥(a) = (−1)|R+\R+(a)| Y α∈R+\R+(a) ξα1/2(γ) − ξα−1/2(γ)2, det (1 − Ad (γ)) |z⊥(a) 1/2 = D(γ) Y α∈R+\R+(a) ξα1/2(γ) − ξα−1/2(γ) (4.12) Y α∈Rre +\Rre+(a) ξα−1/2 k−1 , det 1 − Ad k−1 |z⊥ a(γ) = (−1) |R+(a)\R+(γ)| Y α∈R+(a)\R+(γ) ξα1/2(γ) − ξα−1/2(γ)2.
Proof. Using Theorem 3.18 and the third identity in (3.70), we get (4.13) det (1 − Ad (γ)) |z⊥(a) =
Y
α∈R+\R+(a)
(1 − ξα(γ)) 1 − ξα−1(γ) ,
from which the first equation in (4.12) follows.
By proceeding as in Subsection 3.5, we find that −θ acts on R+\
R+(a) ∪ R+re without fixed points, so that
R+\ R+(a) ∪ Rre+ is even, and so (4.14) (−1)|R+\R+(a)| = (−1)|Rre+\Rre+(a)| .
The same arguments also show that
(4.15) Y
α∈R+\(R+(a)∪Rre+)
ξα1/2(γ) − ξα−1/2(γ)
is a positive number, and also that
(4.16) Y α∈R+\(R+(a)∪Rre+) (1 − ξα(γ)) 1 − ξα−1(γ) = Y α∈R+\(R+(a)∪Rre+) ξ1/2α (γ) − ξ−1/2α (γ)2.
